-x^2-3x+28=-এর সমাধান করুন

$$- x ^ { 2 } - 3 x + 28 =$$

Factor

$$\left(4-x\right)\left(x+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -x^{2}+ax+bx+28. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-28=-28$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -28.

$$1,-28 2,-14 4,-7$$

Calculate the sum for each pair.

$$1-28=-27 2-14=-12 4-7=-3$$

The solution is the pair that gives sum -3.

$$a=4 b=-7$$

Rewrite -x^{2}-3x+28 as \left(-x^{2}+4x\right)+\left(-7x+28\right).

$$\left(-x^{2}+4x\right)+\left(-7x+28\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x+4\right)+7\left(-x+4\right)$$

Factor out common term -x+4 by using distributive property.

$$\left(-x+4\right)\left(x+7\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-x^{2}-3x+28=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 28}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 28}}{2\left(-1\right)}$$

Multiply 4 times 28.

$$x=\frac{-\left(-3\right)±\sqrt{9+112}}{2\left(-1\right)}$$

Add 9 to 112.

$$x=\frac{-\left(-3\right)±\sqrt{121}}{2\left(-1\right)}$$

Take the square root of 121.

$$x=\frac{-\left(-3\right)±11}{2\left(-1\right)}$$

The opposite of -3 is 3.

$$x=\frac{3±11}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{3±11}{-2}$$

Now solve the equation x=\frac{3±11}{-2} when ± is plus. Add 3 to 11.

$$x=\frac{14}{-2}$$

Divide 14 by -2.

$$x=-7$$

Now solve the equation x=\frac{3±11}{-2} when ± is minus. Subtract 11 from 3.

$$x=-\frac{8}{-2}$$

Divide -8 by -2.

$$x=4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and 4 for x_{2}.

$$-x^{2}-3x+28=-\left(x-\left(-7\right)\right)\left(x-4\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$-x^{2}-3x+28=-\left(x+7\right)\left(x-4\right)$$

Evaluate

$$\left(4-x\right)\left(x+7\right)$$